<h2>Problem 198</h2>
<div style="color:#666;font-size:80%;">14 June 2008</div><br />
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<p>A best approximation to a real number <var>x</var> for the denominator bound <var>d</var> is a rational number <var>r</var>/<var>s</var> (in reduced form) with <var>s</var> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <var>d</var>, so that any rational number <var>p</var>/<var>q</var> which is closer to <var>x</var> than <var>r</var>/<var>s</var> has <var>q</var> <img src='images/symbol_gt.gif' width='10' height='10' alt='&gt;' border='0' style='vertical-align:middle;' /> <var>d</var>.</p>

<p>Usually the best approximation to a real number is uniquely determined for all denominator bounds. However, there are some exceptions, e.g. 9/40 has the two best approximations 1/4 and 1/5 for the denominator bound 6.
We shall call a real number <var>x</var> <i>ambiguous</i>, if there is at least one denominator bound for which <var>x</var> possesses two best approximations. Clearly, an ambiguous number is necessarily rational.</p>

<p>How many ambiguous numbers <var>x</var> = <var>p</var>/<var>q</var>,
0 <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> <var>x</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> 1/100, are there whose denominator <var>q</var> does not exceed 10<img src="" style="display:none;" alt="^(" /><sup>8</sup><img src="" style="display:none;" alt=")" />?</p>
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